Definition of an algebra over a monad by using equalities between natural transformations

Question

In the definition of a monad, there are two ways to specify the equations:

1. equalities between natural transformations, or
2. equalities between morphisms

as is done there on Wikipedia.

In the usual definition of an algebra over a monad, equalities are between morphisms (1$^\text{st}$ way) like there. How do you reformulate these equations as equalities between natural transformations (2$^\text{nd}$ way)?

What I tried: I have been looking for the answer in the definition of a left module over a monad because it is a generalization of algebras in bicategories.

Answer 1

A module for a monad $(T,\eta,\mu)$ on $D$ is a functor $F:C\to D$ together with a natural transformation $\alpha:T\circ F\to F$ such that $\alpha\circ\eta=\mathrm{id}_F$ and $\alpha\circ (\mu F)=\alpha\circ (T\alpha).$ Set $C$ to the terminal category to get an ordinary $T$-algebra.